Some applications of logic to feasibility in higher types

In this paper we demonstrate that the class of basic feasible functionals has recursion theoretic properties which naturally generalize the corresponding properties of the class of feasible functions. We also improve the Kapron - Cook result on mashine representation of basic feasible functionals. Our proofs are based on essential applications of logic. We introduce a weak fragment of second order arithmetic with second order variables ranging over functions from N into N which suitably characterizes basic feasible functionals, and show that it is a useful tool for investigating the properties of basic feasible functionals. In particular, we provide an example how one can extract feasible "programs" from mathematical proofs which use non-feasible functionals (like second order polynomials)

Mathematische Logik

[no abstract available

Semantics vs. Syntax vs. Computations Machine Models for Type-2 Polynomial-Time Bounded Functionals (Preliminary Draft)

This paper investigates analogs of the Kreisel-Lacombe-Shoenfield Theorem in the context of the type-2 basic feasible functionals, a.k.a. the Mehlhorn-Cook class of type-2 polynomial-time functionals. We develop a direct, polynomial-time analog of effective operation, where the time bound on computations is modeled after Kapron and Cook\u27s scheme for their basic polynomial-time functionals. We show that (i) if P = NP, these polynomial-time effective operations are strictly more powerful on R (the class of recursive functions) than the basic feasible functions, and (ii) there is an oracle relative to which these polynomial-time effective operations and the basic feasible functionals have the same power on R. We also consider a weaker notion of polynomial-time effective operation where the machines computing these functionals have access to the computations of their functional parameter, but not to its program text. For this version of polynomial-time effective operation, the analog of the Kreisel-Lacombe-Shoenfield is shown to hold-their power matches that of the basic feasible functionals on R

Global semantic typing for inductive and coinductive computing

Inductive and coinductive types are commonly construed as ontological (Church-style) types, denoting canonical data-sets such as natural numbers, lists, and streams. For various purposes, notably the study of programs in the context of global semantics, it is preferable to think of types as semantical properties (Curry-style). Intrinsic theories were introduced in the late 1990s to provide a purely logical framework for reasoning about programs and their semantic types. We extend them here to data given by any combination of inductive and coinductive definitions. This approach is of interest because it fits tightly with syntactic, semantic, and proof theoretic fundamentals of formal logic, with potential applications in implicit computational complexity as well as extraction of programs from proofs. We prove a Canonicity Theorem, showing that the global definition of program typing, via the usual (Tarskian) semantics of first-order logic, agrees with their operational semantics in the intended model. Finally, we show that every intrinsic theory is interpretable in a conservative extension of first-order arithmetic. This means that quantification over infinite data objects does not lead, on its own, to proof-theoretic strength beyond that of Peano Arithmetic. Intrinsic theories are perfectly amenable to formulas-as-types Curry-Howard morphisms, and were used to characterize major computational complexity classes Their extensions described here have similar potential which has already been applied

A Two-Layer Approach to the Computability and Complexity of Real Numbers

We present a new approach to computability of real numbers in which real functions have type-1 representations, which also includes the ability to reason about the complexity of real numbers and functions. We discuss how this allows efficient implementations of exact real numbers and also present a new real number system that is based on it